Simplify and expand the following expression: $ \dfrac{p}{p + 9}-\dfrac{p + 4}{3p - 2} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(p + 9)(3p - 2)$ Multiply the first term by $\dfrac{3p - 2}{3p - 2}$ $ \begin{align*} \dfrac{p}{p + 9} \times \dfrac{3p - 2}{3p - 2} & = \dfrac{(p)(3p - 2)}{(p + 9)(3p - 2)} \\ & = \dfrac{3p^2 - 2p}{(p + 9)(3p - 2)}\end{align*} $ Multiply the second term by $\dfrac{p + 9}{p + 9}$ $ \begin{align*} \dfrac{p + 4}{3p - 2} \times \dfrac{p + 9}{p + 9} & = \dfrac{(p + 4)(p + 9)}{(3p - 2)(p + 9)} \\ & = \dfrac{p^2 + 13p + 36}{(3p - 2)(p + 9)}\end{align*} $ Now we have: $ = \dfrac{3p^2 - 2p}{(p + 9)(3p - 2)} - \dfrac{p^2 + 13p + 36}{(3p - 2)(p + 9)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3p^2 - 2p - (p^2 + 13p + 36)}{(p + 9)(3p - 2)} $ $ = \dfrac{3p^2 - 2p - p^2 - 13p - 36}{(p + 9)(3p - 2)} $ $ = \dfrac{2p^2 - 15p - 36}{(p + 9)(3p - 2)}$ Expand the denominator: $ = \dfrac{2p^2 - 15p - 36}{3p^2 + 25p - 18}$